Beyond the Four-Level Model: Dark and Hot States in Quantum Dots Degrade Photonic Entanglement

Entangled photon pairs are essential for a multitude of quantum photonic applications. To date, the best performing solid-state quantum emitters of entangled photons are semiconductor quantum dots operated around liquid-helium temperatures. To favor the widespread deployment of these sources, it is important to explore and understand their behavior at temperatures accessible with compact Stirling coolers. Here we study the polarization entanglement among photon pairs from the biexciton–exciton cascade in GaAs quantum dots at temperatures up to ∼65 K. We observe entanglement degradation accompanied by changes in decay dynamics, which we ascribe to thermal population and depopulation of hot and dark states in addition to the four levels relevant for photon pair generation. Detailed calculations considering the presence and characteristics of the additional states and phonon-assisted transitions support the interpretation. We expect these results to guide the optimization of quantum dots as sources of highly entangled photons at elevated temperatures.

E ntangled photon pairs have been used to explore the validity of quantum mechanics and some of its least intuitive predictions. 1 Besides being intriguing, entanglement is a key resource to establish correlations among remote locations, to achieve resolution beyond classical capabilities and for quantum information processing. 2−4 In the last decades different methods have been developed to generate entangled photon pairs, 5 such as parametric down-conversion, 6,7 which has led to sources that can be operated in a wide temperature range 8 and also in satellites. 9 However, the stochastic photon generation process leads to an increase of the multipair generation probability and thus to a degradation of entanglement 10,11 when the brightness is increased. In contrast, semiconductor quantum dots (QDs) 12 are quantum emitters exhibiting sub-Poissonian emission characteristics and ultralow multiphoton pair emission probability even at maximum brightness. As a consequence, it may become possible for QDs, e.g., to outperform the secure key rate achievable with probabilistic sources in entanglement-based quantum-keydistribution. 13 In ideal QDs, a polarization-entangled photon pair can be obtained by initializing the system in a biexciton |XX⟩ state, 14,15 which decays back to the crystal ground state |G⟩ via two bright and energy-degenerate excitonic |X H/V ⟩ states following two possible decay paths [inset of Figure  1(b)]. In particular, GaAs QDs obtained via the local droplet etching (LDE) method in an AlGaAs matrix 16−20 have demonstrated excellent performance as sources of single entangled-photon pairs with a fidelity to one of the maximally entangled Bell states as high as 0.98. 21,22 Thus far, the best results have been obtained at cryogenic temperatures, reachable with liquid-helium-based (wet) cryostats or bulky and energy-intensive closed-cycle (dry) cryostats. To achieve further advances with QD light sources, possibly enabling their deployment in space applications with light and energyefficient cryo-coolers, 23 a comprehensive study of entanglement at different operation temperatures T is needed. By using strain-tunable GaAs QDs capable of generating nearly perfectly entangled photons at low temperatures, 21 we investigate the effects produced by increasing T on the entanglement and on the exciton decay dynamics following coherent |XX⟩ excitation. While the multipair emission probability remains low for all investigated temperatures, we find that entanglement degrades for T ≳ 15 K, which is accompanied by a slowed excitonic decay and weak light emission from higher-energy ("hot") excitonic states. To gain insight into this rich evolution, we (i) expand the four-level model for the biexciton−exciton cascade by including hot and dark excitonic states, (ii) address the properties of such additional levels as well as of the corresponding radiative and nonradiative transitions by experiments and 8-band k·p and configuration-interaction (CI) calculations, (iii) model the population dynamics with the Liouville−von Neumann equation with Lindblad terms and rate equations, and (iv) evaluate the two-time correlation functions and degree of entanglement based on density matrix methods. Our calculations reproduce very well the experimental results and provide evidence that both the degradation of entanglement and the changes in decay dynamics for the |XX⟩ → |X H/V ⟩ (shortly XX) and |X H/V ⟩ → |G⟩ (X) transitions can be traced back to the thermal population and depopulation of excited and dark exciton states and to spin mixing.
GaAs QDs grown via the LDE method can be optimized to have |X H/V ⟩ almost degenerate. 16,17 Nevertheless, a finite energy splitting (or fine-structure splitting, FSS) generally remains, leading to a time evolution of the entangled state and consequent entanglement degradation in time-integrated measurements. 24 Therefore, we make use of a piezoelectric strain-tuning device 21,25 to cancel the FSS; see Figure 1(a). For the optical excitation of an individual QD, we use resonant two-photon excitation (TPE) [inset of Figure 1(b)] by tuning the energy E L of a pulsed laser with a 80 MHz repetition rate to half of the difference between the |XX⟩ and |G⟩ states 26,27 and by setting the laser power to obtain the maximum XX intensity (π-pulse conditions). The recorded photoluminescence (PL) spectrum under TPE at T = 64.4 K is shown in Figure 1(b). Besides the XX at 1.5761 eV and X at 1.5799 eV, a weaker line at 1.5836 eV is also visible, which we denote as X* and attribute to a thermally populated excitonic state. Spectra collected at different temperatures, showing also further excited states, can be found in the Supporting Information. 28 To assess the effect of T on the light emission characteristics and entangled photon generation following optical excitation, we performed our study by increasing T stepwise, in a range from 4.4 to 64.4 K. For each T value we canceled the FSS via strain-tuning. First, the g (2) autocorrelation functions of both the XX and X were recorded. In Figure 1(c) a clear broadening of the X histogram peaks is visible at T = 64.4 K compared to low-temperature data. In addition, a slight increase in g (2) (0) (see Figure 3 Supporting Information 28 ) for XX and X from g XX (2) (0) = 0.008(1) to g XX (2) (0) = 0.032(5) and from g X (2) (0) = 0.008(1) to g X (2) (0) = 0.033(3) is visible. A full state tomography was performed to obtain the twoqubit density matrices in polarization space, using a maximum likelihood method. 29 Two representative density matrices for 4.4 and 48.4 K are shown in Figures 1(d) and (e). For higher temperatures, we observe decreasing VV-HH coherence with respect to HH-HH and VV-VV occupations, as well as rising HV and VH elements, indicative of state mixing. We evaluate concurrence and fidelity at every temperature for a time bin of 2 ns, as shown in Figure 1(f). At low temperatures the concurrence [fidelity] is equal to 0.94(1) [0.969(4)], comparable to former investigations. 21,30,31 The degree of entanglement stays approximately constant up to about 15 K and then decreases with increasing temperature. The slight increase in g (2) (0) mentioned above does not explain the observed steep entanglement degradation shown in Figures 1(d−f) since the g (2) (0) values are still in the range typically observed at T = 5 K. 30,32,33 To gain further insights into the origin of the entanglement degradation, we study the decay dynamics of the XX and X  Figure 1(b), we add two dark exciton states |X D ⟩, as well as excited states of the exciton |X*⟩ and biexciton |XX*⟩; see Figure 2(b). In the single-particle picture, the "hot" |XX*⟩ and |X*⟩ states are configurations where the electrons are in the "s-shell" and the holes in the "p-shell". The energy difference ΔE = 3.7 meV between |X⟩ and |X*⟩ is taken from the recorded spectrum in Figure 1(b). Because |X*⟩ consists of an electron and a hole, four spin configurations are possible, resulting in four possible transitions. For purely heavy-hole excitons we would expect two bright and two dark states, similar to the ground-state exciton. In ref 36 a triplet was instead observed, which we ascribe to the high light-hole contribution of almost 40% (with ∼25% of bright admixture) to "p-shell" holes. 36−38 As we are not interested in the detailed population of the excited states, we include |X*⟩ as a single state with multiplicity of four. For the |XX*⟩, we expect two "s-shell" electrons in a singlet state and two holes, one in the "s-shell" and the other in the "p-shell", resulting again in four possible configurations. Since we were not able to unequivocally identify the emission lines associated with |XX*⟩, we assume the same value of 3.7 meV for the |XX⟩ − |XX*⟩ energy separation. Finally for the bright-dark splitting we take a value of 110 μeV, as in ref 36, and a multiplicity of two. We note that excited states with the electron in the "p-shell" are unlikely to be populated in the explored temperature range due to significantly higher energy differences of 15−20 meV according to our calculation (see Supporting Information 28 ). States with holes in higher-energy shells play instead a role for T ≳ 40 K (see below) but are omitted from our model for the sake of simplicity. With reference to the levels shown in Figure 2(b) we now focus on the radiative recombinations (solid lines) and nonradiative phonon-assisted transitions (dashed lines) and their rates. Different from the dominant XX and X emission lines, which are characterized by the rates γ X and γ XX , the recombination rate γ X * of |X*⟩ is relatively weak because of the different envelope function symmetry for the electron and hole, but clearly visible at high temperature [ Figure 1(b)], under nonresonant excitation, 36 and in PL-excitation measurements. 32,39 For radiative recombinations involving the same single-particle states we assume the same values for the corresponding rates. As an example, the recombination of |XX*⟩ leaving the system in a ground-state exciton |X H/V ⟩ (|X D ⟩) takes place with a rate γ XXd 1 * (γ XXd D * ) with γ XXd 1 * = γ XXd D * = γ X *, since in all cases we have a "p-shell" hole recombining with an "s-shell" electron. Further, the rate γ XXd 2 * for the |XX*⟩ recombination leaving the system in the |X*⟩ state is assumed to be equal to γ X since an "s-shell" electron recombines with an "s-shell" hole.
For the phonon-assisted transitions we assume a Tdependence that is determined by the expected phonon number according to the Bose−Einstein distribution: where γ PH 0 = 1 ns −1 is the phonon-assisted relaxation rate at low temperature, estimated from simulations (see section "Phonon-assisted relaxation" of the Supporting Information 28 ) and fully consistent with the slow relaxation previously reported in refs 32 and 34 for similar QDs. |X*⟩ is populated via the phonon-mediated rate 4γ PH * , where the factor 4 in this and  Table 1.

Nano Letters pubs.acs.org/NanoLett
Letter other rates corresponds to the state multiplicity discussed above, with To justify the assumption of equal phonon-assisted rates for transitions involving states with different spin configurations and to fully understand the entanglement degradation observed in Figure 1(f), it is important to note that the highly mixed character of excited hole levels implies that the spin projection along the growth axis is not a good quantum number for the hot states. As a consequence, phonon-assisted relaxations are effectively not spin-conserving (see Supporting Information). This is in good agreement with PL-excitation measurements, where the |X*⟩ was excited resonantly and equal intensities for the X V and X H transitions independent of laser polarization were observed after relaxation. Calculations also confirm this almost "spin-agnostic" relaxation from all four |X*⟩ to all bright and dark |X⟩ states with at most 40% difference in rates (see Supporting Information), which justifies taking them equal for the sake of simplicity. The same approach is followed for the transitions involving |XX*⟩. Note that spin-flips between bright and dark exciton states are neglected since they are expected to occur at time scales in the order of μs. 35 All rates used in our model and the corresponding origins are summarized in Table 1.
Solving the rate equations for the presented system and convolving the obtained time evolutions with a measured instrument response function result in the decay traces shown in the lower panels of Figure 2(a). The simulation reproduces both the initial acceleration of the decay observed for the XX and X�due to population loss through thermal excitation of the hot states�and the pronounced slow decay of the X signal�due to the repopulation of the bright |X V/H ⟩ states. As in the experiment, an increasing T mostly affects the X dynamics. From the model, the slowed X decay is mostly produced by the slow phonon-assisted relaxation rate γ PH 0 compared to γ X combined with the state multiplicity of the |X*⟩ state, which thus acts as a reservoir slowly feeding the population of the |X V/H ⟩ states. We notice that, for T ≳ 40 K, the measured X decay is still slower than the predicted decay, which we attribute to additional phonon-assisted population of higher-energy states, that are not taken into account in the presented model (see Supporting Information).
We now turn to the effect of the thermally induced processes described above on the degree of entanglement of XX-X photon pairs. To this end, numerical simulations are performed, solving the corresponding Liouville−von Neumann equation. In a first step, the photonic two-qubit density matrix is theoretically calculated based on polarization-resolved, timeintegrated two-time correlation functions, 40 modeling the experimental measurements. Afterward the simulated concurrence is directly evaluated from the obtained photonic density matrices. The result of these simulations is shown in Figure 1(f) in light blue for temperatures from 0 to 70 K. Note that the simulations do not predict a unity concurrence for low temperatures even for zero FSS, since the TPE sets a limit to the obtainable degree of entanglement due to a dynamic Stark shift of one exciton level induced by the excitation itself. 40 The simulated result reproduces well the concurrence plateau at lower temperatures followed by a decrease starting around 16 K. Again, theory and experiment are in good agreement up to T = 40 K. For higher temperatures, the theory predicts a slower degradation compared to the experiment, consistent to additional phonon-assisted excitation channels.
In addition to the concurrence calculations, we use the decay model shown in Figure 2(b) to compute the photonic twoqubit density matrices and compare them with the experimentally reconstructed matrices, where mixing and decoherence emerge with increasing T; see Figure 1(e). In Figure 3(a) a representative histogram of the difference between the detection times of X and XX photons at T = 32.4 K is shown, displaying on the right side of the peaks coincidences that arise from the slow X decay. Coincidences within the chosen time bin of 500 ps (black dashed lines), corresponding approximately to our detector resolution are summed up and compared with the average area of the side peaks in the same interval. Figure 3(b) shows the measured and simulated real part of the density matrices for this time bin. The leftmost 2D diagrams in the gray and orange boxes in Figure 3(c) correspond to the 3D representations in Figure  3(b). Next, we begin to shift the time bin to higher time delays, indicated by the blue arrows in Figure 3(a) and above Figure  3(c). The resulting density matrices for a time delay up to 1050 ps are shown in panel (c). The gray box shows the measurement, the orange box the density matrices obtained from corresponding simulations. In order to mimic the timefiltering analysis theoretically, only photon pairs with a delay time in the respective interval/time bin are considered in the time-integrated correlation functions; cf., Supporting Information. Both experiment and theory show increasing state mixing (rise of HV-HV and VH-VH elements) and decoherence (drop of HH-VV and VV-HH elements) with increasing time delay. In turn, this finding is consistent with our dynamic model, in which the detection events producing the "tail" in the coincidence histograms stem from thermal cycling among levels, i.e. the occupation of hot and dark states at elevated temperatures followed by bright exciton repopulation with no spin memory.
In summary, we have investigated the effect of rising operation temperature on the quality of the polarization entanglement of photon pairs generated by the biexciton− exciton decay cascade in a single GaAs QD tuned to have negligible excitonic fine-structure splitting. By performing fullstate tomography including time-filtering and lifetime measurements under resonant optical excitation as well as dedicated calculations, we ascribe both the entanglement degradation Values taken from the PL measurements following TPE excitation at 4.4 K. b Value estimated from the comparison between the results of the rate equation model and PL intensities of the X* line as well as from the k·p and CI simulations (see Supporting Information 28 ). c Values estimated from the k·p and CI simulations.

Nano Letters pubs.acs.org/NanoLett
Letter and the changes in decay dynamics to thermal cycling among the desired |XX⟩ and |X H/V ⟩ states and "undesired" hot and dark states, which are connected to the former by phononassisted transitions leading to spin scattering and decoherence. In turn, the spin-agnostic character of the transitions is traced back to the high valence-band mixing in the excited states of the employed QDs. From the achieved understanding one could envision that an increased energy splitting ΔE can substantially extend the plateau of high concurrence at lower temperatures up to 40 K, reachable with available Stirling coolers. 23,41 Since the excited states will be less populated for larger energy splittings, the impact of thermal cycling will be reduced. An increasing ΔE is also expected to lead to a reduction of hole mixing in the excited states benefiting the preservation of high entanglement. Consequently, we anticipate that QDs capable of generating highly entangled photons with relaxed operation-temperature requirements can be obtained by slightly reducing the QD size. For GaAs QDs, this can be simply achieved by reducing the amount of GaAs filling. 42 Recent work 50 on (presumably strongly confining) InGaAs QDs shows indeed the persistence of high levels of entanglement beyond 90 K. However, along with a change in QD size one must consider also pure dephasing 43 due to the deformation potential coupling to longitudinal acoustic (LA) phonons. 44−46 This mechanism has been shown to reduce the concurrence by enhancing offresonant single-photon transitions and decoherence. 47 Although the effects of pure dephasing seem to be insignificant for the QD studied in this work, they become more relevant with decreasing QD size as the coupling to LA phonons becomes more effective. 48,49 . In conclusion, finding the optimal structural properties of QDs capable of emitting highly entangled photons at elevated temperatures will need further understanding of various effects and their impact.

■ ASSOCIATED CONTENT Data Availability Statement
The data of this study is available from the corresponding author upon request.